Set- theore(c methods in model theory

Transcription

1 Set- theore(c methods in model theory Jouko Väänänen Amsterdam, Helsinki 1

2 Models i.e. structures Rela(onal structure (M,R,...). A set with rela(ons, func(ons and constants. Par(al orders, trees, linear orders, ladces, groups, semigroups, fields, monoids, graphs, hypergraphs, directed graphs. 2

3 Models and topology A countable model is a point in 2 ω (mod ). A model of size κ is a point in 2 κ (mod ). Proper(es of models subsets of 2 κ. Isomorphism of models:``analy(c subset of 2 κ x 2 κ. 3

4 How to iden(fy a structure? Relevant even for finite structures. Can infinite structures be classified by invariants? 4

5 Shelah s Main Gap M any structure. The first order theory of M is either of the two types: Structure Case: All uncountable models can be characterized in terms of dimension- like invariants. Non- structure case: In every uncountable cardinality there are non- isomorphic models that are extremly difficult to dis(nguish from each other by means of invariants. 5

6 Non- structure The Main Gap 2 κ Number of models 1 1 Structure Size of the model κ 6

7 The program To analyze further the non- structure case. We replace isomorphism by a game. We develop topology of 2 κ. 7

8 Gameclock for EF 4 (A,B) The non- isomorphism player starts The isomorphism player responds Two players: The non- isomorphism player and the isomorphism player. 8

9 Approxima(ng isomorphism M,N countable (graphs, posets,...) M = N The non- isomorphism player wins the EF game of length ω with the enumera(on strategy τ T(M,N)=the countable tree of plays against τ, where the isomorphism player has not lost yet. T(M,N) has no infinite branches, well- founded 9

10 Approxima(ng isomorphism (contd.) T(M,N) has a rank α<ω 1. σ M =sup a,b {rank(t((m,a),(m,b)) : (M,a) = M,b)} Scof rank of M. Scof ranks put countable models into a hierarchy, calibrated by countable ordinals. The orbit of M is a Borel subset of ω ω. 60 s and 70 s: Scof, Vaught invariant topology 90 s and 00 s: Kechris, Hjorth, Louveau: Borel equivalence rela(ons 10

11 Game with a clock The isomorphism player loses the EF game of length ω, but maybe she can win if the non- isomorphism player is forced to obey a clock. 11

12 The Non- isomorphism non- isomorphism player goes up player makes the a clock- tree move 12

13 The clock gives a chance Although the isomorphism player loses the EF game of length ω, she wins the game which has T(M,N) as the clock. T(M,N)=the tree of plays against τ, where the isomorphism player has not lost yet. 13

14 A well- founded clock The tree B α of descending sequences of elements of α is the canonical well- founded tree of rank α 14

15 For countable M and N: TFAE: M N The isomorphism player wins the EF game clocked by B α for all α<ω 1. TFAE: M N The isomorphism player wins the EF game clocked by B α for some α<ω 1 such that the non- isomorphism player wins with clock B α+1 α+1 α = = 15

16 An ordering of trees, mo(vated by games T T if there is f:t T such that x< T y f(x)< T f(y). If T and T do not have infinite branches, then T T iff rank(t) rank(t ). Fact: T T iff II wins a comparison game on T and T. 16

17 T T ranks game clocks If T T then a game clocked by T is easier for the isomorphism player harder for the non- isomorphism player than the same game clocked by T. 17

18 Isomorphism player wins + Isomorphism player wins Non- isomorphism player wins + Non- isomorphism player wins Isomorphism player wins + Non- isomorphism player wins < 18

19 There are incomparable trees (Todorcevic) There are incomparable Aronszajn trees. A tree is a bofleneck if it is comparable with every other tree. (Mekler- V., Todorcevic- V.) It is consistent that there are no non- trivial boflenecks. (Todorcevic) PFA coherent Aronszajn trees are all comparable, and there is a canonical family of coherent Aronszajn trees that are boflenecks in the class of trees of size ℵ 1 (Aronszajn trees). 19

20 The structure of trees of size and height ℵ 1 under ω 1 Aronszajn trees well- founded trees i.e. ordinals <ω 2 20

21 A ``successor operator on trees T a tree σt = the tree of ascending chains in T T< σt σb α B α+1 21

22 The uncountable case M,N of size κ (graphs, posets,...) M = N The non- isomorphism player wins the EF game of length κ with the enumera(on strategy τ. T(M,N)=the tree of plays against τ, where the isomorphism player has not lost yet. T(M,N) has no branches of length κ, ``bounded. The cardinality of T(M,N) is κ <κ. 22

23 The uncountable case For M and N of cardinality κ TFAE: M N The isomorphism player wins the EF game clocked by T for all trees T w/o κ- branches, T 2 κ<κ. The non- isomorphism player loses the EF game clocked by T for all trees T w/o κ- branches, T κ <κ. 23

24 Watershed For M and N of cardinality κ TFAE: = M N The isomorphism player wins the EF game clocked by K for some tree K w/o κ- branches, K 2 κ<κ, but does not win the game clocked by σk The non- isomorphism player does not win the EF game clocked by S for some tree S w/o κ- branches, S κ <κ, but wins if clocked by σs. 24

25 σs σk S K 25

26 Non- determinacy of the EF game Determinacy of the EF game of length ω 1 in the class of models of size ℵ 2 is equiconsistent with the existence of a weakly compact cardinal. (HyDnen- Shelah- V.) 26

27 Generalized Baire space ω 1 ω 1, models of size ℵ 1 G δ - topology. ω 1 - metrizable, ω 1 - addi(ve. meager ( α<ω1 A α, A α nowhere dense), Baire Category Theorem holds: B α dense open α<ω 1 B α. dense set of con(nuum size. Sikorski, Todorcevic, Shelah, Juhasz &Weiss, κ κ, models of size κ λ κ, κ=cof(λ), models of size λ, which are unions of chains of length κ of smaller models. 27

28 Descrip(ve Set Theory in ω 1 ω 1 A set A ω 1 ω 1 is analy(c if it is the projec(on of a closed set ω 1 ω 1 x ω 1 ω 1. Equivalently, there is a tree T ω 1 <ω 1 x ω 1 <ω 1 such that f A iff T(f) has an uncountable branch, where T(f)={g(α) : (g(α),f(α)) T} and g(α)=(g (β)) β<α. 28

29 A Covering Theorem Every co- analy(c subset A of ω 1 ω 1 is covered by canonical sets B T, T a tree w/o uncountable branches, such that every analy(c subset of A is covered by some B T. CH implies the sets B T are analy(c and the trees T are of size ℵ 1. 29

30 Covering Theorem under CH a co- analy(c set analy(c subsets 30

31 Proof Suppose A is co- analy(c and B A is analy(c. f A iff T(f) has an uncountable branch. f B iff S(f) has no uncountable branches. Let T be the tree of (f(α),g(α),h(α)) where g(α) T(f) and h(α) S(f). If f B, there is an uncountable branch h in S(f). Let F(g(α))= (f(α),g(α),h(α)). This is an order preserving mapping T(f) T 31

32 Proof contd. So T(f) T Let A T ={f A : T(f) T }. Then B A T. We have proved the Covering Theorem: If A is co- analy(c, then A is the union of sets A T such that if B is any analy(c set A, then there is a tree T w/o uncountable branches such that B A T. CH implies each A T is analy(c. 32

33 Souslin- Kleene, separa(on Souslin- Kleene: If A is analy(c co- analy(c, then A=A T for some T w/o uncountable branches. Separa(on: If A and B are disjoint analy(c sets in ω 1 ω 1, then there is a set C=(- B) T which separates A and B. 33

34 Luzin Separa(on Theorem? Borel means closure of open under complements and unions of length ω 1. (Shelah- V.) Assume CH. There are disjoint analy(c sets which cannot be separated by a Borel set. Assume CH+MA. Any two disjoint analy(c sets of expansions of (ω 1,<) can be separated by a Borel set. (Halko, Mekler, Shelah, V.) CUB is not Borel, but ``CUB is analy(c co- analy(c is independent of ZFC+CH, as is ``the orbit of the free group of ℵ 1 generators is analy(c co- analy(c. 34

35 Definable trees and/or models? (J. Steel) Assuming large cardinals, If T R <ω 1 is in L(R), then ``T has an uncountable branch is forcing absolute. If M and N are in L(R) and their universe is ω 1, then M N is absolute with respect to forcing that preserves ω 1. 35

36 The analogy Ordinals No descending chains Finite Successor ordinal Game clock Comparison of ordinals Undefinability of well- order Trees No uncountable branches Countable The tree of all chains of a tree Clock tree Order- preseving mappings Undefinability of having an uncountable branch Baire space ω ω Generalized Baire space ω 1 ω 1 Analy(c union of countable ordinals is countable Analy(c union of trees with no uncountable branches is a tree with no uncountable branches 36

37 2 κ Number of models 1 1 Size of the model κ 37

38 Degrees of under CH = First proved by HyDnen and Tuuri 38

39 Cardinal invariants about trees U(κ) Universality Property: There is a family of size κ of trees of size and height ℵ 1 w/o branches of length ω 1 such that every such tree is one in the family. B(κ) Boundedness Property: Every family of size < κ of trees of size and height ℵ 1 w/o branches of length ω 1 has a tree which is each one in the family. C(κ) Covering Property: Every co- analy(c subset A of ω 1 ω 1 is covered by κ analy(c sets, such that every analy(c subset of A is covered by one of them. 39

40 Universal set Bounded set 40

41 Cardinal invariants about trees U(κ) Universality Property B(κ) Boundedness Property C(κ) Covering Property (U(κ)&B(λ)) C(κ)&λ κ, (B(κ)&λ<κ) C(λ) U(κ) & B(κ) is consistent with κ anything between ℵ 2 and 2 ℵ 1. (Mekler- V. 1993) U(κ + ) & B(κ + ) if ℵ 1 replaced by a singular strong limit, of cof ω. (Dzamonja- V. 2008) 41

42 A recent result of Shelah There are structures M and N such that The cardinality of M and N is ℵ 1. For all α<ω 1, the isomorphism player wins the EF game of length α. M and N are non- isomorphic. Note: CH not assumed. 42

43 Summary In the non- structure case we can get models that are very close to being isomorphic in the sense that the non- isomorphism player does not win even if he is given a large clock tree. the isomorphism player wins in large clock trees. We need to understand the structure of trees befer. 43

44 Thank you! 44